Global geometry and topology of spacelike stationary surfaces in $\mathbb{R}^4_1$

Details Global geometry and topology of spacelike stationary surfaces in $\mathbb{R}^4_1$ Global geometry and topology of spacelike stationary surfaces in $\mathbb{R}^4_1$

2011-03-24

arxiv.org/abs/1103.4700v5

Mathematics

Differential Geometry

35 pages. Compared to previous versions, in this complete manuscript we add three sections on construction of various examples

Scientific paper

For spacelike stationary (i.e. zero mean curvature) surfaces in 4-dimensional Lorentz space one can naturally introduce two Gauss maps and Weierstrass representation. In this paper we investigate their global geometry systematically. The total Gaussian curvature is related with the surface topology as well as the indices of the so-called good singular ends by a generalized Jorge-Meeks formula. On the other hand, as shown by a family of counter-examples to Osserman's theorem, finite total curvature no longer implies that Gauss maps extend to the ends. Interesting examples include the generalization of the classical catenoids, helicoids, the Enneper surface, and Jorge-Meeks' $k$-noids. Any of them could be embedded in $\mathbb{R}^4_1$, showing a sharp contrast with the case of $\mathbb{R}^3$.

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